This norm is also called the supremum norm, the Chebyshev norm, or the infinity norm. The name "uniform norm" derives from the fact that a sequence of functions converges to f under the metric derived from the uniform norm if and only if converges to uniformly.[1]

If we allow unbounded functions, this formula does not yield a norm or metric in a strict sense, although the obtained so-called extended metric still allows one to define a topology on the function space in question.

where D is the domain of f (and the integral amounts to a sum if D is a discrete set).

The binary function

is then a metric on the space of all bounded functions (and, obviously, any of its subsets) on a particular domain. A sequence { fn : n = 1, 2, 3, ... } converges uniformly to a function f if and only if

We can define closed sets and closures of sets with respect to this metric topology; closed sets in the uniform norm are sometimes called uniformly closed and closures uniform closures. The uniform closure of a set of functions A is the space of all functions that can be approximated by a sequence of uniformly-converging functions on A. For instance, one restatement of the Stone–Weierstrass theorem is that the set of all continuous functions on is the uniform closure of the set of polynomials on .